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%% lecture19.tex
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%% Started on  Thu Jan  5 08:26:18 2012 alex
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\exercises
\begin{xca}
For finite cell complexes $X$ and $Y$ show that $\chi(X\times
Y)=\chi(X)\times\chi(Y)$.
\end{xca}
\begin{xca}
If a finite cell complex $X$ is union of subcomplexes $A$ and $B$ show that
\begin{equation}
\chi(X)=\chi(A)+\chi(B)-\chi(A\cap B).
\end{equation}
\end{xca}
\begin{xca}
Let us consider a function $\phi$ assigning a real number to
every finite cell complex. Let us assume that $\phi$ is a
topological invariant and $\phi(X)=\phi(A)+\phi(X/A)$ if $A$ is a
subcomplex of $X$. Prove that $\phi$ can be represented in the
form $\phi(X) = const(\chi(X)-1)$.
\end{xca}
\begin{xca}
Let us suppose that a closed orientable surface $M_g$ of genus
$g$ (a sphere with $g$ handles) is an $n$-sheeted covering space
of $M_h$ (of sphere with $h$ handles). Prove that $g=n(h-1)+1$.
\end{xca}
